This leaves us with imprimitive groups having blocks of size \(q\)  and
containing no transitive subgroups having blocks of size \(p\) .
The case \(q=2\)  was settled in [1]}, [2]}.
As mentioned in Section 
we will construct in Section 
a family of imprimitive groups with blocks of size \(q\) 
having intersection density equal to \(q\) , thus
disproving Conjecture REF (iii).
In Example REF  below we give the smallest counter-example to this
conjecture, a group of degree \(3\cdot 11\)  with intersection density 3.
This group, however, is not part of the family
of groups from Theorem REF 
where the smallest group occurs for \(q=3\) 
and \(p=13\) .
In summary, in order to obtain
a complete characterization of intersection densities
of transitive groups of degree \(pq\) , simply primitive groups
from the above mentioned list and imprimitive groups
with blocks of size \(q\)  will have to be addressed.
